Geodesic
Superdiffusions and positive solutions of non-linear partial differential equations
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 59 (2004) no. 1, pp. 147-157

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By using super-Brownian motion, all positive solutions of the non-linear differential equation $\Delta u=u^\alpha$ with $1\alpha\leqslant 2$ in a bounded smooth domain $E$ are characterized by their (fine) traces on the boundary. This solves a problem posed by the author a few years ago. The special case $\alpha=2$ was treated by B. Mselati in 2002. 
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     author = {E. B. Dynkin},
     title = {Superdiffusions and positive solutions of non-linear partial differential equations},
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E. B. Dynkin. Superdiffusions and positive solutions of non-linear partial differential equations. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 59 (2004) no. 1, pp. 147-157. http://geodesic.mathdoc.fr/item/RM_2004_59_1_a9/